My neighbour threatened me. I know this forum is full help.
Yesterday I was working on SONAR. Didn't notice the time - it was 1:20 AM.
Two of them came and threatened to break my "Sound System"
I managed to buy 2 pcs of Behringer 2030A.
 
But now I have to buy a headphone. Not having much of cash. What can be my options? [Also confused about closed back / open / semi-open etc..]
 
What do I do?
 
Regards
Snehankur

If you have to work on headphones at night a good choice would be ATH M50's, they go on sale sometimes. You might also look at getting a VRM box to simulate rooms and speakers through your headphones. Just always keep an eye on the volume levels because it's easy to get carried away with loud levels through headphones and you can damage your ears. Lots of breaks required.

Yeah always a good idea to respect your neighbors at night. It can be very stressful when you have to get up early for work and there's a thump thump thump from a kick drum next door. I've been driven nuts like this before, banging on the wall bleary eyed and yelling like a miserable old fart, lol. So I'm acutely aware of people being asleep when I want to make music at night.

Move!     No, I am just kidding, of course. Headphones, to me, while important don't have to be the best and most expensive!!! Some people will argue that point. From my own experiences I have learned that MOST finished mixs are 90% of the time done on speaker monitors.
Get the best phones you can afford that are comfortable for long periods of time and use them for those times when it will be a nuisance to anyone to hear you. Just keep in mind that what you need to do under the phones should be tasks concerning choosing fx, adjusting tempos, correcting wrong notes, adjusting loops, any technical things. Phones can get you in the neighborhood of a good mix but my experiences taught me that speakers are crucial to finished mixs. Mixing should be done with the speakers and checked on several different types of speakers. All this can be done in the waking hours with the hopes your not disturbing people around you too much. Phones as I use them are for tracking and for zeroing in on trouble issues.
And, PLEASE, watch those levels while under the phones. I have said before to google ear fatigue and understand what you risk when listening thru either speakers or phones!

Normally I remain careful. Yesterday I didn't notice the time. I get up around 4.30-5.00AM and I can only get this 9.30 PM - .. for my music. BTW what about Closed-back, open, semi open type of headphone?

I use closed back when tracking live. But, I have found open back to work better for me when mixing.
If you plan on mixing with headphones, get a decent pair and use the Sonarworks Headphone calibration plugin. That made a world of difference in my mixes.
Just my $.02 YMMV
 

Open back shouldn't bother your neighbors (your head would explode first).  I like the sound of my AKG 240s I got fairly cheap - they sounded better than the close-backeds in the same $100 price range I tried.
 
Another solution - I had a friend crashing at a studio in the bad side of San Antonio while they worked on their first album.  One night late there was a heavy knocking on the door.  Paul yelled through the bolted door, and whomever was on the other side started yelling to open up, they knew the owner, yada yada yada.  The owner had wisely left a .45, and Paul chambered a round - a very loud sound.  There was an "oh man" and they left quickly.

Hi. I have and still use an AKG K240 (I think they are about 13 years old, but still as new). They are quite neutral, but I have problems when wearing them for a long time (many hours). They make my ears sweat 
I just bought a set of Beyerdynamics DT770 (80 Ohm), and I can now work for (too many) hours without a sweat (pun intended). 
All the best.

I have used a pair of Beyerdynamic DT 770 Pros for around 3 years for mixing when neighbours are around (which is often). FWIW, people who've listened to the resultant mixes on their own various devices  have said, "Good mix!" They seem to quite neutral. Also very comfortable to wear for a long time, even on my fat head. 

 
Since they sometimes come up, I suggest staying away from the Sennheiser HD280 Pros (which I own and are about 100$).  Being closed they have to be tight and so become uncomfortable after an hour or so.  Closed also makes them bass heavy.  They're fantastic phones for listening in noisy environments like on a plane, and the overall sound is great, but I wouldn't buy them for mixing.  AKG and Beyerdynamics are the classic studio phones which I'd look at if I were buying new ones.

Snehankur
My neighbour threatened me. I know this forum is full help.
Yesterday I was working on SONAR. Didn't notice the time - it was 1:20 AM.
Two of them came and threatened to break my "Sound System"
I managed to buy 2 pcs of Behringer 2030A.
 
But now I have to buy a headphone. Not having much of cash. What can be my options? [Also confused about closed back / open / semi-open etc..]
 
What do I do?
 
Regards
Snehankur

Maybe you are having it too loud to being with.  I never like to abuse my hearing.